Properties

Label 21T138
Degree $21$
Order $7348320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3^6.(C_2\times S_7)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 138);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $138$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^6.(C_2\times S_7)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,16,2,18,3,17)(4,20,5,21,6,19)(7,9)(10,12)(13,15), (1,17,5,7,13,2,16,6,9,14)(3,18,4,8,15)(10,21,12,20,11,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5040$:  $S_7$
$10080$:  $S_7\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: $S_7$

Low degree siblings

42T2539, 42T2540, 42T2541

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

There are 118 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $7348320=2^{5} \cdot 3^{8} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  7348320.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);